 1. Introduction
 2. Experimental layout
 3. Data processing
 4. Flux density comparison
 5. Transport of the intensity after the grating
 6. SHADOW ray tracing and phase space results
 7. van Cittert–Zernike theorem
 8. Comparison with beamline data
 9. Fourreflection – 8.500 keV horizontal
 10. Direct – 8.500 keV
 11. Results at 8.200 keV
 12. Summary
 References
 1. Introduction
 2. Experimental layout
 3. Data processing
 4. Flux density comparison
 5. Transport of the intensity after the grating
 6. SHADOW ray tracing and phase space results
 7. van Cittert–Zernike theorem
 8. Comparison with beamline data
 9. Fourreflection – 8.500 keV horizontal
 10. Direct – 8.500 keV
 11. Results at 8.200 keV
 12. Summary
 References
research papers
Increased spatial coherence length from an asymmetric crystal reflection at grazing exit
^{a}Advanced Photon Source, Argonne National Laboratory, Argonne, IL 60439, USA, and ^{b}Ecopulse, Inc., 7844 Vervain Ct, Springfield, VA 22152, USA
^{*}Correspondence email: atm@anl.gov
Coherent Xray imaging is an active field at synchrotron sources. The images rely on the available coherent
over a limited field of view. At many synchrotron beamlines a doublecrystal monochromator (DCM) is employed in a standard nondispersive arrangement. For coherent diffraction imaging it is advantageous to increase the available field of view by increasing the spatial coherence length (SCL) of a beam exiting such a DCM. Here, Talbot interferometry data together with raytracing simulations for a (+ − − +) fourreflection experimental arrangement are presented, wherein the first two reflections are in the DCM and the final fourth reflection is asymmetric at grazing exit. Analyses of the interferometry data combined with the simulations show that compared with the beam exiting the DCM a gain of 76% in the SCL was achieved, albeit with a factor of 20 reduction in which may not be a severe penalty at a synchrotron beamline. Previous efforts reported in the literature to increase the SCL that employed asymmetric crystal diffraction at grazing incidence are also discussed. A much reduced SCL is found presently in simulations wherein the same asymmetric crystal is set for grazing incidence instead of grazing exit. In addition, the present study is compared and contrasted with two other means of increasing the SCL. These are (i) focusing the beam onto an aperture to act as a secondary source, and (ii) allowing the beam to propagate in vacuum an additional distance along the beamline.1. Introduction
Coherent Xray imaging is an active field at synchrotron sources that employs the available coherent ; Jacobsen, 2020). The field of view that can be imaged is limited by the spatial coherence length (SCL). For these studies it is often advantageous to have a field of view as large as possible so that larger samples can be imaged coherently. The present work details measurements by which the SCL was increased over that available directly from the beamline doublecrystal monochromator (DCM) at beamline 1BM, a bending magnet beamline at the Advanced Photon Source (APS), USA.
over a limited field of view (Nugent, 2010We report simulations that demonstrate that the spatial coherence length can be increased significantly for an asymmetric Si 111 Bragg reflection at grazing exit. A fourreflection (+ − − +) diffraction arrangement was employed for which the first two reflections were in the DCM. The beam from the DCM was first diffracted dispersively from a symmetric Si(111) crystal and then diffracted from an asymmetric Si 111 Bragg reflection at grazing exit. The asymmetry angle built into the crystal was 13.1°. Talbot measurements for the coherence length were obtained at APS beamline 1BM (Marathe et al., 2014; Macrander et al., 2016) at 8.500 keV. A value of 39.8 µm was measured for the SCL. For the beam directly exiting the DCM a value of 22.6 µm was measured. In addition, extensive ray tracing (Rebuffi & Sanchez del Rio, 2016, 2017) simulations simulate the data well. At grazing exit the effective source is much reduced in size and was found to deviate from a Gaussian profile. The effective source was found to be only 3.2 µm FWHM, a quite small value. Raytracing simulations with the effective source are presented. The effects of were included automatically in the ray tracing. We report that the grazing exit geometry can indeed increase the SCL significantly over that for a beam exiting a DCM.
Asymmetric Bragg diffraction at grazing incidence instead of at grazing exit has previously been considered to improve the spatial coherence of synchrotron beams. Collimation is achieved in a grazing incidence geometry, and since the SCL is inversely proportional to a divergence, albeit at a specific point in the object field, a longer SCL might be achieved. In general, there is the complication of chromatic aberration, that is et al. (1995a) report that the grazing incidence geometry is severely plagued by such chromatic aberration. The work by Ishikawa (1988) reports an SCL of 220 µm, albeit for a final monochromatic beam with a divergence of only 0.04 µrad achieved after two sequential asymmetric reflections at grazing incidence. The sample in their case was a wedged crystal studied in a nondispersive Laue geometry. This nondispersive arrangement cleverly countered the effect of chromatic aberration. However, a required experimental arrangement that mitigates the effects of chromatic aberration complicates the general usefulness of the double grazing incidence geometry reported by Ishikawa. See Appendix A for comments on a grazing incidence geometry for the fourth crystal in our experimental arrangement. See Appendix B for a comparison of the present method with other methods to increase the SCL.
arising from a finite bandwidth which increases the output divergence from an asymmetric crystal. BrauerSomewhat counterintuitively, a reverse or grazing exit geometry was found presently to yield a significantly longer SCL than was available directly from the DCM. We note that a grazing exit geometry has been studied previously with the intent to improve the coherent et al., 1995b). The present study was instead focused on simply increasing the spatial coherence length for the purpose of coherent diffraction imaging studies. This was found to be possible, with a gain of 76% over the SCL of the beam directly exiting the DCM, albeit with a factor of 20 reduction in density.
through a pinhole with a diameter of a few micrometres (Brauer2. Experimental layout
Fig. 1 shows a schematic layout in side view at beamline 1BM of the APS that was used for the present measurements. The whitebeam radiation from the bending magnet was apertured by slits to 3.0 mm vertically by 3.0 mm horizontally. The two silicon crystals in the DCM were set for the 111 reflection in a classic (+ −) geometry to produce a beam parallel to the main axis of the beamline. The first crystal was water cooled (Lang et al., 1999). The physical distance between the bending magnet source and the grating was 35.5 m. However, for the beam directly from the DCM the source distance obtained from the Talbot interferometry in the vertical plane was 6 m shorter, and this could be simulated with a heat bump on the first crystal of the DCM (see below). The heat load of 1 W was such that a heat bump on the first crystal was expected. This heat bump was approximately 20 nm in height as estimated from wiggler results for a 1 W heat load at beamline ID19 at the European Synchrotron Radiation Facility (Rutishauser et al., 2013). Relevant changes made after the published description of the beamline (Macrander et al., 2016) were: (i) removal of Xray beryllium windows upstream of the DCM in conjunction with implementation of differential pumping such that the vacuum space in the DCM was contiguous with that in the storage ring, and (ii) installation of polished beryllium windows of 500 µm total thickness placed downstream of the DCM.
Downstream of these windows, two more silicon crystals were configured in a sequential arrangement (Stoupin et al., 2016). The first of these was a 111 oriented silicon crystal set to diffract dispersively in a (+ − −) geometry as shown in Fig. 1. The second downstream crystal (the fourth crystal overall) was a silicon crystal with 111 planes oriented at 13.1° with respect to the surface. This final crystal was set to diffract at grazing exit. The exit angle at 8.5 keV was 0.41° and at 8.2 keV was 0.88°. The overall diffraction geometry for the fourcrystal diffraction arrangement was (+ − − +).
A benefit of this fourcrystal optical arrangement is a reduced bandpass. The bandpass after the third crystal was 0.8 eV FWHM and 0.2 eV after the fourth crystal. The beam incident on the fourth crystal corresponds to a longitudinal coherence length, λ^{2}/(Δλ), equal to 2.8 µm. This value exceeds the extinction length of the asymmetric reflection which is 0.7 µm. We conclude that the longitudinal coherence length is not limiting presently, a conclusion further supported by the agreement we achieved between our SHADOW simulations and our Talbot interferometry measurements.
A Talbot interferometer based on a checkerboard pure phase grating was located downstream at a distance of 3.32 m from the asymmetrically diffracting crystal. The period of the checkerboard grating was 4.8 µm (Shi et al., 2022). Alternating square areas of the grating were gold plated to a thickness of 1.62 µm. This provided a phase shift of π at the design energy for the grating of 8.0 keV. Very small phase shift differences at 8.2 keV and 8.5 keV did not affect the present analyses. Similarly, transmission factors differing from the design value of 0.52 at 8.0 keV up to 0.57 at 8.5 keV did not affect the present results, a finding attributable to the procedure of applying the secondorder Fourier transform factor normalized to the zeroth order (Shi et al., 2022). A LuAG:Ce Xray scintillator which was imaged with 10× lens onto an Andor Neo CMOS detector was situated on a linear slide. The pixel size of the detector was 6.5 µm, which when combined with the 10× lens yielded a resolution of 0.65 µm for the Xray patterns incident on the scintillator.
For image data the linear stage was scanned downstream along the beamline axis, labeled the zaxis, out to 900 mm in either 2 or 4 mm steps. Data were obtained not only for the fourreflection geometry but also for a beam exiting directly from the DCM, that is, for a tworeflection geometry. This served both to demonstrate the increased SCL attainable compared with the tworeflection geometry and to compare densities, since the fourreflection geometry comes at the cost of a reduced density.
3. Data processing
The image data were processed by wavepy2 (WavePy2.0, https://github.com/APSXSDOPTGroup/wavepy2). The processing yielded points as a function of z for fringe visibilities along both vertical and horizontal directions. The visibility oscillates with a decreasing maximum on a length scale equal to oneeighth of the Talbot distance. The Talbot distance is given by 2p^{2}/λ, where p is the period (= 4.8 µm). The Talbot distance at 8.500 keV was 316 mm. The function by which the maxima decrease is known in the literature as the complex coherence factor (CCF) (Pfeiffer et al., 2005), where the actual applied factor is the absolute value of a complex quantity. For an incoherent source the CCF can be calculated by means of the van Cittert–Zernike theorem (Goodman, 1985). The processing of the measured data proceeded by taking the Fourier transform and equating the amplitude of the second order, normalized by the zeroth order, to a visibility. As well, wavepy2 yields a fit to the visibility with a Gaussian function. However, a Gaussian function was not assumed presently. The CCF was instead calculated from an effective source obtained by ray tracing to which the van Cittert–Zernike theorem was then applied to yield a simulated CCF.
4. comparison
Fig. 2 shows crosscuts of detector images for both the fourreflection and the tworeflection geometries. For the fourreflection case the counting time was 120 s per image, whereas the counting time for the tworeflection geometry was 6 s. The images were obtained at z values of 1/16 of the Talbot distance. At 1/16 of the Talbot distance, and odd multiples thereof, the secondorder Fourier coefficient of a crosscut profile is at a maximum. This is conveniently seen by examining the socalled `Talbot carpet' of a π phase grating (Weitkamp et al., 2006). The profiles in Fig. 2 are for six periods of the grating. For each period there are two peaks. One peak is weaker owing to absorption in the gold squares.
For the fourreflection geometry, the wavelength–angle phase space region on a Dumond diagram is reduced in comparison with that of tworeflection geometry (Dumond, 1937), and as a result the throughput is reduced. The reduction presently can be quantified from the profiles shown in Fig. 2. The comparison shows that densities for the fourreflection geometry was 5% of that for the tworeflection geometry.
5. Transport of the intensity after the grating
Talbot interferometer data were simulated by transporting the electric field, and thus the intensity, as a function of z, first transmitted through the grating and then propagated from the grating to the scintillator. The transmission and propagation was carried via equation (1) (Momose et al., 2003; Guigay, 1971),
Here, y is the vertical transverse coordinate, x is the horizontal transverse coordinate, ϕ is a phase shift of π at the checkerboard pads and zero between pads, and λ is the Xray wavelength. This equation was applied, and results for the twodimensional partial Talbot effect were found to be in agreement with prior results (Weitkamp et al., 2006; Zanette, 2006). That is, there was agreement for the Talbot carpet. The resulting intensities as a function of z were then Fourier transformed in two dimensions and the secondorder Fourier coefficient was computed along both the vertical and horizontal directions. This produced the result shown in Fig. 3 which applies to a spherical incident wave. Here the zscale was stretched to account for a spherical incident wave instead of a plane wave (Weitkamp et al., 2006). The results in Fig. 3 do not account for the partial coherence of the beam incident on the grating. Ray tracing was carried out in order to simulate such partial coherence.
6. SHADOW ray tracing and phase space results
As part of the present study, extensive ray tracing was carried out via the ShadowOui program available from the OrAnge SYnchrotron Suite (OASYS) (Rebuffi & Sanchez del Rio, 2016, 2017). The second crystal in the DCM was detuned to reduce the throughput by 50% in order to suppress Si 333 harmonic radiation. This setting mimicked the procedure used during the measurements. Ray tracing was carried out downstream of the asymmetric crystal reaching to the grating.
The raytracing results were used to simulate an effective source as shown in Fig. 4 which shows (vertical distance)/(vertical divergence) phase spaces both at the grating in the lower panel as well as at the deduced effective source located upstream of the grating and shown in the upper panel. These results show the divergence of rays at specific vertical positions. A symmetric phase space diagram, such as would be used to represent the bending magnet source, for example, when propagated downstream spreads to larger vertical distances but retains the same spread in divergence. So, for example, rays with the largest divergence spread to the largest vertical position. With the raytracing result at the grating, it is a simple matter to reverse this propagation and determine at which distance upstream the phase space diagram becomes symmetric. This is how we obtained the effective source position. The effective source determined in this way is shown in the upper panel in Fig. 4 and was found to occur at 3.32 m upstream, very near the asymmetric crystal.
The van Cittert–Zernike theorem was then applied to the effective source.
7. van Cittert–Zernike theorem
The CCF, denoted in equation (2) as μ, between two points in the observation plane separated by Δy vertically and by Δx horizontally was calculated as per the van Cittert–Zernike theorem (Goodman, 1985) for an incoherent source. The applicable equation, to within a phase factor, is
Here, I(ξ, η) is the intensity in the source plane at transverse coordinates ξ and η, and D is the distance between the source and the grating. Since the grating was a checkerboard, the CCF that equals the absolute value of μ could be obtained as a function of both vertical and horizontal directions. We first apply this equation at the grating and note that it is a Fourier transform with frequencies given by f = Δy/λD for the vertical direction and Δx/λD for the horizontal direction.
Interference occurs downstream of the grating, which can be viewed as a beamsplitter and which relates the Δy and z scales as (Pfeiffer et al., 2005)
This procedure can also be described as projecting the source profile (Weitkamp et al., 2006) or as the application of a transfer function (Shi et al., 2022). The net result is that the CCF can be related to z settings for the measurements via the Fourier transform frequency, that is, by applying z = pDf/2.
The effective source shown in the upper panel of Fig. 4 was projected onto the vertical yaxis to yield the profile shown in the upper panel of Fig. 5. The raytracing results, shown as points, were fit to a polynomial function. This function was then Fourier transformed, and via the frequency the CCF function shown in the lower panel of Fig. 5 was obtained.
To determine a value for the SCL, the z value for a decrease of exp(−1/2) = 0.606, denoted as z*, was applied in equation (3). This procedure mimics that for a Gaussian as given in
Here, a value of exp(−1/2) is obtained for l_{c} = Δy*, and Δy* is obtained from equation (3) with application of z*. The net result is an SCL value given by 2λz*/p. A value of 38.9 µm was obtained for the SCL in the case of Fig. 5.
8. Comparison with beamline data
The effect of partial coherence was simulated by means of the CCF shown in Fig. 5. The product of this function and the calculated visibility shown in Fig. 3 is shown in Fig. 6. This figure also shows the data points that were obtained by applying wavepy2 to the beamline data. The overlay of the simulation and the data points demonstrate a good agreement.
9. Fourreflection – 8.500 keV horizontal
The same procedures were followed for the horizontal direction. The results for the fourreflection geometry at 8.500 keV in the horizontal direction are shown in Fig. 7. The distance D between the effective source and the grating was found by ray tracing to be 35.5 m. That is, the effective source was the bending magnet itself. This is expected for a heat bump limited to the vertical plane. A value of 9.7 µm was found for the SCL in this case.
10. Direct – 8.500 keV
Measurements and simulations were also made for a beam directly exiting the DCM. In the vertical plane a value of D equal to 28.8 m was found, which is shorter than the distance to the bending magnet. By introducing a heat bump of 20 nm on the first crystal in the DCM, a value of 29 m for D was also obtained by means of ray tracing. The results are shown in Fig. 8. A value of 22.6 µm was found for the SCL in the vertical direction. We note that this reveals a gain of 76% for the SCL of the fourreflection case over the tworeflection case, albeit with a factor of 20 reduction in Results in the horizontal plane are shown in Fig. 9. A value of D equal to 35.5 m was found for both the data and the simulations. This distance corresponds to the distance from the bending magnet, confirming again that the heat bump was only effective in the vertical plane.
11. Results at 8.200 keV
A full set of measurements were also made at 8.200 keV for which the exit angle is 0.88°, that is, roughly twice that at 8.500 keV. The Talbot interferometry measurements could also be well simulated with the 50% detuning setting implemented at the start of zscans during the data taking. The FWHM of the effective source was 8.0 µm, a considerably larger value than was obtained at 8.500 keV, namely 3.2 µm. This larger FWHM resulted in an SCL value of 19.7 µm, a value considerably reduced from the 38.9 µm value for 8.500 keV. The results at 8.200 keV are summarized in Table 1 along with the results at 8.500 keV that were presented above. Notable here is the reasonable agreement between all values of D and between all values of the SCL in the horizontal direction. Of particular note is the reduced value of 23.4 µm for the SCL in the vertical direction for the fourreflection configuration at 8.200 keV. We conclude that the fourreflection geometry must be at quite low grazing exit conditions to be effective in increasing the SCL.

12. Summary
In summary, in comparison with a beam exiting a DCM, the spatial coherence length was found to have an increased value for a (+ − − +) fourreflection geometry in which the fourth reflection is asymmetric and configured at grazing exit. Detailed Talbot interferometry results obtained at beamline 1BM at the Advanced Photon Source with a checkerboard grating are presented together with extensive raytracing simulations at 8.500 keV and 8.200 keV. Results in both the vertical and the horizontal directions are reported not only for the fourreflection case but also for the beam directly exiting the DCM. An effective source located very near the asymmetric crystal resulted from the ray tracing, leading to a determination of the CCF which caused the amplitude of Talbot fringes to decrease with increasing distance between the grating and detector. The CCF was determined without the assumption that it was a Gaussian function. A value of 38.9 µm was obtained for the spatial coherence length at 8.500 keV for a Si 111 reflection with an asymmetry angle of 13.1°. This corresponded to an exit angle of 0.41°. This spatial coherence length was 76% larger than that obtained for the beam that exited the DCM.
APPENDIX A
Comments on grazing exit versus grazing incidence
At grazing incidence the beam footprint and exit beam was obtained at the grating position at 8.500 keV. We note that this phase space result is in stark contrast to that for the grazing exit geometry shown in the lower panel of Fig. 4. The divergence at the center of the grating evident in Fig. 10 is α = 25 µrad FWHM, and, when applied to (Brauer et al., 1995b)
can be increased over that for a symmetric reflection. Concomitant is a decreased exit divergence, that is, the exit beam is more collimated. The grazing incidence geometry was also simulated presently by means of ray tracing. The (vertical distance)/(vertical divergence) phase space result shown in Fig. 10yields a value of only 3.0 µm for the SCL. A significant aspect of the present work is that one must consider not only the usual 2d Dumond phase space of wavelength and divergence but also include position as a third axis. With this included by means of SHADOW, the grazing exit geometry does produce a wide overall vertical exit divergence, as shown in the lower panel of Fig. 4, but the divergence at a single vertical position is much reduced to 3 µrad. As per the van Cittert–Zernike theorem, the divergence that sets the spatial coherence length is the divergence at a single point in the object field. For the grazing incidence geometry, although the overall divergence is roughly an order of magnitude smaller, a vertical divergence of 25 µrad pertains at all points in the object field which results in a significantly shorter coherence length compared with the grazing exit case.
APPENDIX B
Comparison with other ways to increase the spatial coherence length
In this Appendix we compare our method with two other means to increase the spatial coherence length. These two means are: (i) use a focusing optic to focus on a secondary source, and (ii) simply allow the beam to propagate in vacuum. SHADOW simulation results are compared and contrasted with the fourreflection case as summarized in Table 2. In both cases (i) and (ii), the only crystals in the beam path were in the DCM.

For case (i) we simulated a phasetype Fresnel zone plate (FZP) focused on a circular aperture. This aperture was located at the same downstream position as was the grazing exit crystal, 32.2 m, and the distance from the aperture to the object plane, i.e. the plane of the grating, was also set to be the same, 3.32 m. The diameter of the FZP was taken to be 1.00 mm, which is near a realistic maximum for a FZP as set by lithography fabrication methods. The outermost zone width was 175 nm, and the focal length was 120 cm. For the purpose of making the comparisons summarized in Table 2, we treated all beams as having a Gaussian shape. This resulted in a slightly larger (by 4 µm) SCL of 43 µm for the present fourbounce grazing exit arrangement. In the case of the FZP, the waist at the focus was 34 µm FWHM. A value of 34 µm was set for the diameter of the secondary source aperture. The upstream distance of the effective source, D, was determined in SHADOW to be 3.26 m, and the effective source size value for σ was 1.48 µm. For comparison purposes the SCL applicable in the case of a Gaussian source, namely, λD/2πσ (Marathe et al., 2014), is listed in Table 2. This value was 51 µm, a value somewhat comparable with the grazing exit fourreflection case presented above. We note that the parameters for the FZP and secondary source were chosen to make SCL come out roughly comparable with the grazing exit case. The final column in Table 2 lists a simulated intensity. The intensity found in all three cases summarized in Table 2 was for one million rays emanating from the bending magnet in SHADOW simulations. The intensity for the FZP plus a secondary source case was found to be only 5 as compared with a value of 263 in the case of the grazing exit crystal presented above. This amounts to reduction of a factor of 53 in intensity, which is a relative disadvantage. There is the additional disadvantage of having to block scattering from the edges of the secondary source.
For case (ii) we simulated a beam that was allowed to propagate an additional distance from the DCM. For comparison purposes we set this additional distance to be such that the SCL, 37 µm, was again roughly comparable with the value obtained for the grazing exit crystal case presented above. In this case the σ value for the effective source size was 31 µm, and the effective source distance, D, was 49.7 m. As listed in Table 1 for the twobeam vertical case a value of D equal to 29 m was measured in the case of a beam coming directly from the bending magnet which resulted in an SCL of 23 µm. As summarized in Table 2, an additional propagated distance of 21 m is required to increase this value to 37 µm, albeit with an improved intensity by a factor of 3.5 over that of the grazing exit crystal case. The disadvantage of the pure propagation case is the need for a beamline that is longer by 21 m.
Acknowledgements
We acknowledge the contributions of Elina Kasman for crystal finishing and of Jun Qian for measurements of surface roughness. We are indebted to Mark Sutton for related comments. This work was supported by the US Department of Energy, Office of Basic Energy Sciences under contract No. DEAC0206CH11357.
Funding information
The following funding is acknowledged: US Department of Energy, Office of Science (contract No. DEAC0206CH11357).
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